Photonic crystals having a skeleton structure

ABSTRACT

The invention relates to a class of photonic crystals that are similar to the known inverse opals while being characterized by so far not known band gaps or larger pseudo band gaps, especially between the 5 th  and 6 th  band and/or between the 8 th  and 9 th  band. The invention further relates to a method for producing said photonic crystals and to the use thereof as larger resonators, matrices for optical guides, opalescent pigments, beam splitters, spectral filters or as components of such devices.

The invention relates to a class of photonic crystals which are similarto the known inverse opals, but until now have had unknown bandgaps orgreater pseudo bandgaps, in particular between the fifth and sixthbands, and/or between the eighth and ninth bands. The inventionfurthermore relates to a method for production of photonic crystals andtheir use as a laser resonator, a matrix for optical waveguides, anopalescent pigment, a beam splitter, a spectral filter or as a componentof such apparatuses.

BACKGROUND TO THE INVENTION

Photonic crystals are materials in which the refractive index is variedperiodically in three dimensions, with lattice constants in the regionof the light wavelength being of particular interest. The periodicvariation of the material is applied to the light waves which propagatein these media (see, for example, J. D. Joannopoulos, R. D. Meade, J. N.Winn, Photonic Crystals: Molding the flow of light. Princeton UniversityPress, 1995. It has been found that these periodically modulated wavescannot be produced for any frequency for a selected direction.Particularly if the wavelength of these waves virtually matches thenetwork plane separations in the photonic crystal, the propagation ishighly modified by multiple scatter, and its frequency is decreased orincreased. This results in gaps in the frequency scale, in which thereis no mathematical solution for the propagation problem ofelectromagnetic waves in the periodic material. Depending on whetherthese gaps occur for all directions of electromagnetic waves, or only ina restricted range of directions, these frequency gaps are referred toas bandgaps or pseudo bandgaps. Their calculation requires theprocessing of the wave equation for the electromagnetic field, for whichpurpose methods have been developed which are used in a similar mannerto the Schrodinger equation in a periodic potential (J. D. Joannopoulos,R. D. Meade, J. N. Winn, Photonic Crystals: Molding the flow of light.Princeton University Press, 1995; K. Busch, S. John, Phys. Rev. E 58(1998) 3896; S. G. Johnson, J. D. Joannopoulos, Optics Express 8 (2000)173).

Photonic crystals are of major interest for use in optical components,such as those which are used in the communications industry. Inparticular, materials with large optical bandgaps allow novel opticalfunctionalities (F. Marlow, Nachrichten Chem. [Information Chem.] 49(2001) 1018).

Photonic crystals may be produced on the basis of spherical packages.However, dense packaging of spheres with a high refractive index neverhas a complete bandgap, although its inverse structure, with closelyarranged spherical cavities in a material with a high refractive index,may have. A production method has been derived from this in whichoriginal spherical packaging is used as a molding (negative mold) forthe inverse structure, which will be referred to in the following textas the remaining volume structure (RVS) when it completely fills themolding (Y. A. Vlasov, N. Yao, D. J. Norris, Adv. Mater. 11 (1999) 165;A. Zakhidov et al. U.S. Pat. No. 6,261,469). Furthermore, productionmethods have been found for shell structures, in which the molding isfilled with a layer of a material (J. E. Wijnhoven, W. L. Vos, Science,281 (1998) 802).

Remaining volume structures are predominantly concave, since they arederived from a convex mold (spherical packaging). This characteristiccan be described as follows by the mean radius of curvature <{overscore(R)}> of a structure whose total surface area is A:$\left\langle R \right\rangle = {\frac{1}{A}\underset{A}{\int\int}{\mathbb{d}A}{\overset{\_}{R}(x)}}$${2/{\overset{\_}{R}(x)}} = {{1/R_{1}} + {1/R_{2}}}$

In this case, R₁ and R₂ are the two extreme radii of curvature of thesurface at the point x. By definition, they should be positive if thecenter of the circle of curvature is located in the dense material.Thus, for example, a polystyrene opal composed of spheres whose radiusis R_(ps) has a mean radius of curvature of <R>=R_(ps) (that is to say apositive <R>), and a corresponding ideal RVS (inverse opal) has anegative <R> of <R>=−R_(ps). Structures with a positive <R> are referredto as being predominantly convex, while those with a negative <R> arereferred to as being predominantly concave.

However, predominantly concave structures are subject to a range ofproblems, since structures such as these generally have sharp edges(that is to say relatively small regions with an extremely smallpositive radius of curvature). These edges connect the concave surfaceelements. The behavior of the electromagnetic fields on these edges isextremely complicated. This is evident in calculation difficulties(convergence problems) and in a strong dependency on the precise edgeshape, which restricts the direct control options for the opticalcharacteristics. Furthermore, the implementation options for suchstructures are restricted, since they generally require complete porefillings (in the original opal) as well as very precise control of theexact edge shape.

FIGURE DESCRIPTION

FIG. 1: Band structure for a model system composed of cylinders whichconnect the center points of the octahedral and tetrahedral cavities ofa hexagonally dense spherical packaging (with fcc lattices) to oneanother. The frequency is expressed in the unit c/a, where c is thespeed of light in a vacuum and a is the edge length of the usual cubicunit cell (with four times the volume of the primitive cell) of the fcclattice. The wave vector k varies within the Brillouin zone from Xthrough U, L, Λ, x, W, K back to Λ. In comparison to standardrepresentations, such as that in J. D. Joannopoulos, R. D. Meade, J. N.Winn in Photonic Crystals: Molding the flow of light, PrincetonUniversity Press, 1995, page 80, the link between the K point and the Λpoint was additionally considered.

FIG. 2: Scanning electron microscope record (acceleration voltage 25 kV,40 000 times electron microscope magnification) of a skeleton structure.The connecting pieces are cylindrical and have a cylinder radius ofabout 0.06 a.

DESCRIPTION OF THE INVENTION

The invention is based on the object of providing photonic crystalshaving predominantly convex structures, as well as a method for theirproduction which, on the basis of opal structures, allows the variationof the structure parameters, in particular of the cylinder thickness,and which allows the reproducible synthesis of photonic crystals with abroader application range.

A first aspect of the invention relates to a photonic crystal whosestructure is topologically equivalent to the inverse structure of apredominantly convex molding, characterized in that this crystal

-   -   has a predominantly convex structure, and    -   has a bandgap or pseudo bandgap between the fifth and sixth        bands, and/or    -   has a bandgap or pseudo bandgap between the eighth and ninth        bands,        with at least one bandgap or pseudo bandgap being greater than        that of the inverse structure, which is composed of the same        material as the photonic crystal, of the predominantly convex        molding.

A second aspect of the invention relates to a method for production of aphotonic crystal, based on a predominantly convex molding, comprisingthe following steps:

-   -   (A) penetration of a matrix precursor into the cavities in the        convex molding;    -   (B) conversion of the matrix precursor to the matrix former;    -   (C) redistribution of the matrix precursor/matrix former which        is located in the cavities and/or of their intermediate stages        while maintaining the topology;    -   (D) removal of the molding.

A third aspect of the invention relates to a photonic crystal which canbe obtained by the method described above.

A fourth aspect of the invention relates to use of the photonic crystalaccording to the invention as a laser resonator, a matrix for opticalwaveguides, an opalescent pigment, a beam splitter or a spectral filter,or as a component of the apparatuses mentioned above.

Preferred embodiments of the invention can be found in the dependentclaims.

The photonic crystals according to the invention will be referred to asskeleton structures, since they can be formed from cylindrical ordeformed cylinders or similar individual parts. These individual partsare convex structures, which are held together only by small concavefillets.

For the production of the structures according to the invention, it issignificant that incomplete filling of the molding enlarges thebandgaps. Furthermore, when using suitable production methods, parts ofthese structures may mathematically overlap the molding if the finalphase for production of the skeleton structures takes place at the sametime as the removal of the molding.

The basic predominantly convex moldings, which are used as a “negativemold” for the photonic structures according to the invention, have anopal structure. According to the methods which are known from the priorart for the production of opals, for example sedimentation, the moldingscan be obtained using a large number of materials. The critical factoris that the materials that are used can be shaped to formcorrespondingly small spheres. Polymers and amorphous inorganic oxideshave been found to be particularly suitable, which are chosen from thegroup comprising polystyrene, polymethylmethacrylate (PMMA),polydivinylbenzene, poly(styrene-co-divinylbenzene), melamine resins andsilicon dioxide.

Preferred substances which form the structures according to theinvention (therefore referred to as matrix formers) are oxides,semiconductors, metals and polymers, which are available in the form ofmatrix precursors, preferably in solution. Suitable matrix precursorscomprise at least one compound which is chosen from the groupcomprising:

-   -   (i) Metal alkoxides with the general formula M^(n+)(OR)_(n),        where R is a branched or unbranched hydrocarbon group with 1 to        12, preferably 2 to 8, and particularly preferably 3 to 4,        carbon atoms and M is a metal which is chosen from the groups        IIb, IIIa, IIIb, IVa, IVb and VIIIb in the periodic table of the        elements;    -   (ii) Metal halides or nitrates with the general formula        M^(n+)(X⁻)_(n), where X is a halide ion chosen from F⁻, Cl⁻, Br⁻        and I⁻ or a nitrate ion (NO₃ ⁻), and M is a metal which is        chosen from the groups IIb, IIIa, IIIb, IVa, IVb and VIIIb in        the periodic table of the elements.

Examples of compounds such as these are titanium isopropoxide, aluminumchloride, aluminum nitrate, iron(III) chloride and iron(III) nitrate.

Suitable solvents or additives for the compounds mentioned above arepreferably alcohols or their mixtures, which are chosen from the groupcomprising methanol, ethanol, 1-propanol or 2-propanol and 1-, 2- ortert-butanol. The use of water as a solvent or additive, either on itsown or in a mixture with the solvents -mentioned above, may beadvantageous. In the case of liquid precursors, the use of a solvent oradditive can often be completely dispensed with. The flowingcharacteristics of the precursor and the resultant structure parametersare dependent on the solvent or the additive.

The conversion of the matrix precursors to matrix formers is carried outafter their penetration into the cavities in the molding by calcination,condensation, hydrolysis, oxidation, reduction or drying, orcombinations of the reactions mentioned above.

The precise reaction conditions which are required for conversion of thematrix precursors to the matrix formers are dependent on the nature ofthe chosen matrix precursors. For example, the use of metal alkoxides asprecursors for conversion by hydrolysis and condensation may thereforerequire contact with the moisture in the air.

However, the critical factor in all cases is that the cavities in themolding are not filled completely by shrinkage of the matrixprecursors/formers, such shrinkage taking place during the conversion ofthe matrix precursor to the matrix former.

This shrinkage itself allows deliberate redistribution of the matrixprecursors/formers into defined volume segments of the cavities in themolding, forming the desired skeleton structure.

The critical factor for the achievable range of structure parameters isin many cases that the redistribution of the matrix precursor (C) andthe removal of the molding (D) take place simultaneously.

In this case, gel-like intermediate stages of the inverse structure inthe pores of the opal are preferably used which occur, for example as aresult of condensation, during the conversion of the matrix precursorsto the matrix formers.

Steps (B) and (C) may be carried out simultaneously, particularly whenusing gel-like intermediate stages such as these.

Another option is to use other known production processes for inverseopals (for example A. Zakhidov et al., U.S. Pat. No. 6,261,469), inwhich case an inverse opal of lower density is produced first of all,which is then subjected to a subsequent shrinkage or heat-treatmentprocess, which changes the form of the individual structure elements,while maintaining the topology of the structure. The use of etchingprocesses is also one possible way to subsequently form the inverseopal.

The removal of the molding (C) from the skeleton structure can becarried out by calcination, etching or dissolving.

Depending on the respective molding and matrix precursors/formers, thecalcination process is carried out at temperatures from 450 to 700° C.,preferably 500 to 650° C., and particularly preferably 550 to 600° C.within a time period of 2 to 12 h, preferably 4 to 10 h, andparticularly preferably 5 to 8 h. In this case, it has been found to beadvantageous for the actual calcination process to be preceded by aheating-up phase with a heating rate of 0.8 to 10° C./min, preferably 2to 8° C./min, and particularly preferably 5 to 6° C., and for coolingdown to be carried out at a cooling-down rate of 1 to 15° C./min,preferably 4 to 12° C./min, and particularly preferably 8 to 10° C./min.The calcination process may be carried out in a large number ofdifferent ovens. Suitable ovens are described by the prior art, and havebeen known to those skilled in the art for a long time.

The removal of the moldings by etching or dissolving is preferablycarried out when the moldings which have been used are composed ofmaterials with high thermal stability. For example, a molding composedof silicon dioxide, which cannot be removed by calcination owing to itsvery high thermal stability, can be removed from the skeleton structurewith the aid of hydrofluoric acid (HF). However, this is dependent onthe matrix formers not being attacked by hydrofluoric acid. The photoniccrystals which are produced by the method according to the invention arecharacterized in that these crystals

-   -   have a predominantly convex structure, and    -   have a bandgap or pseudo bandgap between the fifth and sixth        bands, and/or    -   have a bandgap or pseudo bandgap between the eighth and ninth        bands.

The chemical composition of the photonic crystals according to theinvention is admittedly similar to that of known RVS or shellstructures, but they have a considerably different three-dimensionalform.

The crystals according to the invention, which are produced bypreferably incomplete filling of the cavities in a predominantly convexmolding (for example a spherical arrangement), have at least onecomplete bandgap or a pseudo bandgap. A bandgap/pseudo bandgap is inthis case larger than the bandgap or pseudo bandgap of the inversestructure, which is composed of the same material as the photoniccrystal, of the predominantly convex molding.

In a further embodiment according to the invention, the photoniccrystals have two or more bandgaps at the same time.

The bandgaps were verified by means of model calculations. Thecalculations were carried out with the aid of the MIT Photonic Bands(MPB) Software (available as freeware at http://ab-initio.mit.edu/mpb),which is known from the prior art (S. G. Johnson and J. D. Joannopoulosin Optics Express 8 (2000) 173). Typically, 10 bands were calculatedwith the parameters grid-size (16 16 16), mesh-size 7 and tolerance10⁻⁷.

Topologically, the skeleton structures of the photon crystals accordingto the invention are the same as the known inverse structures of opalsor sintered opals (“topologically equivalent”), if this is RVS orvariations of it (for example with rounded edges). In consequence, (a)the lattice type is the same, (b) the existence of links (windows)between cavities does not differ, and (c) the existence of links betweenthe structure crossings is unchanged. Differences in the skeletonstructures and in the RVS occur particularly in the form of linksbetween the structure crossings (a circular cross section or a crosssection similar to a circle in the skeleton structures), and in the formof the structure crossings.

Accumulations of dense material (matrix formers) in the largest cavitiesin the molding, for example in the octahedral, tetrahedral and possiblyother intermediate spaces in spherical packages which provide the shapeare referred to as structure crossings. These intermediate spaces mustbe formed by at least four spheres.

Cavities in the skeleton structures are in this case preferably intendedto mean cavities which are filled with air, although they may also befilled with a material with a low refractive index (lower than that ofthe structure).

The structure is the totality of the accumulations of dense material(matrix formers).

Model calculations on skeleton structures show that a bandgap occursbetween the fifth and sixth bands instead of or in addition to the knownbandgap between the eighth and ninth bands (“5-6 materials” or “5-6/8-9materials”, respectively). In this case, skeleton structures were usedwhich are formed from cylinders and which connect the center points ofthe octahedral and tetrahedral cavities in an opal with an fcc structure(cubic surface-centered lattice) to one another. The bandgap between thefifth and sixth bands occurs above a refractive index contrast n (ratioof the refractive index of the structure to the refractive index of thecavities) of 2.9 in optimized structures (see FIG. 1). This ischaracterized by a maximum of the fifth band at the gamma point, and bya minimum of the sixth band within the Brillouin zone at about k=0.82k(K) when, for example, a skeleton structure composed of cylinders withan optimized radius is used. In this case, k(K) is the quasi impulse atthe K point of the Brillouin zone (for a definition of the K point andgamma point, see O. Madelung, Festkörpertheorie [Solid body theory],Springer Berlin, 1972, page 87). Until now, this bandgap has neitherbeen predicted nor found for RVS nor for a shell structure.

Furthermore, a number of bandgaps below the tenth band (that is to sayat relatively low frequencies) occur simultaneously in the fcc latticeskeleton structures which are composed of cylinders. At the same time asthe bandgap between the fifth and sixth bands, the bandgap between theeighth and ninth bands is opened for a structure with a cylinder radiusof r_(zy)=0.0986 a above about n=3.15. In this case, a is the edgelength of the normal Cartesian unit cell of the fcc lattice, in whichthe primitive unit vectors of the fcc lattice coincide with thehalf-diagonal on the surface (see, for example, Ch. Kittel, Einführungin die Festkörperphysik [Introduction to solid body physics], OldenbourgVerlag, Munich 1999, page 14).

Furthermore, an enlargement of the bandgap was found between the eighthand ninth bands when parts of a skeleton structure are added to an RVSand, in the process, replace parts of the RVS. In consequence, forexample, a bandgap of 6.7% of the mid-frequency of the bandgap can beachieved for the refractive index contrast of n=3.4.

These characteristics open up new application perspectives as photonicbandgap materials for optical circuits and opalescent pigments. Forexample, lasers with a low threshold energy, optical fiber connectionswith extremely small possible radii of curvature, optical beam splittersand components for spectral filtering can be produced, or can beproduced better, on the basis of these photonic crystals, making use ofthe large and more easily achievable bandgaps. The simultaneouslyoccurring bandgaps can be used for simultaneous handling of differentfrequency ranges, for example of two telecommunications windows, in saidcomponents. Furthermore, they allow lasers or similar components inwhich two luminescent species are used, and their luminescence is ineach case suppressed by a different bandgap.

The present invention will be explained in more detail using thefollowing example, although it is not restricted to this example.

EXAMPLE

An inverse TiO₂ opal was produced with the aid of a polystyrene opal (PSopal). This was done using all of the chemicals in the purity assupplied by the manufacturer. As the first step, a PS opal was producedfrom a dilute suspension of PS particles (Microparticles Company,diameter: 270 nanometers, concentration 1% by weight) by slowly dryingat room temperature (approximately 2 weeks in a covered Petri dish).Pieces of the PS opal of about 1-3 mm³ were then subjected to aprecursor solution for 10 minutes to 15 days, which led to infiltrationof the precursor solution. The precursor solution was typically composedof 80% by volume of titanium isopropoxide (Ti(O-i-C₃H₇)₄, Merck Company)and 20% by volume of ethanol (Merck Company). After the infiltration,the saturated opal pieces were subjected to the environmental air for atleast 1 hour (typically several days), in order to allow a reaction withthe moisture in the air. The resultant composite material was, finally,calcined in air at 450-700° C.

In the chosen drying conditions (after the precursor infiltration), theresultant samples have a layer structure in the 1-3 mm³ sample pieces.Scanning electron microscope examinations (acceleration voltage 25 kV,40 000 times electron microscope magnification) of the samplesexternally virtually always showed an approximately 1-5 μm thick skin ofTiO₂ without any regular structures in the size range about 10nanometers, which may contain isolated pores. This is adjacent to atransitional layer with a thickness of between 1 μm and 50 μm, in whichinverse opal structures -with a different structure (RVS, shellstructures and skeletons) occur. In this case, fcc lattices, inparticular, are observed. The core of the sample particles virtuallyentirely comprises a skeleton structure, however (see FIG. 2). In thiscase, the links between the structure crossings are virtuallycylindrical, windows formed from these links are polygonal (similar toquadrilaterals, pentagons or hexagons), and there is scarcely anyincrease in the density of structure crossings in comparison to thecylindrical links. Cylinder radii between 0.04 a and 0.12 a wereobtained. This corresponds to positive mean radii of curvature ofbetween 0.08 a and 0.24 a. The edge length a of the conventional unitcell was between 250 and 360 nanometers. On the basis of modelcalculations, it was possible to verify pseudo bandgaps between thefifth and sixth bands for the structures, with these pseudo bandgapsbeing larger than in the topologically equivalent RVS. The calculationswere carried out using the MIT Photonic Bands (MPB) Software (availableat http://ab-initio.mit.edu/mpb), which is known from the prior art (S.G. Johnson and J. D. Joannopoulos in Optics Express 8 (2000) 173).Typically, 10 bands were calculated, with the parameters grid-size (1616 16), mesh-size 7 and tolerance 10-7.

1. A photonic crystal, whose structure is topologically equivalent tothe inverse structure of a predominately convex molding, wherein saidcrystal: has a predominately convex structure, and has a bandgap orpseudo bandgap between a fifth and sixth bands thereof, and/or has abandgap or pseudo bandgap between an eighth and ninth bands thereof,with at least one bandgap or pseudo bandgap being greater than that ofthe inverse structure of the predominately convex molding, which inversestructure is composed of the same material as the photonic crystal.
 2. Amethod for production of a photonic crystal, said photonic crystal beingbased on a predominately convex molding, said method comprising thefollowing steps: (A) penetrating a matrix precursor into cavities in theconvex molding; (B) converting the matrix precursor to a matrix former;(C) redistributing the matrix precursor/matrix former which is locatedin the cavities and/or of their intermediate stages while maintainingthe topology; (D) removing the molding.
 3. Method as claimed in claim 2,wherein steps (C) and (D) are carried out simultaneously.
 4. The methodas claimed in claim 2, wherein the matrix former which is introduceddoes not completely fill the cavities in the molding.
 5. The method asclaimed in claim 2, wherein the molding (C) is removed by calcination,etching or dissolving.
 6. The method as claimed in claim 2, wherein theredistributing (B) is carried out by shrinking during a calcination,drying and/or condensation of the matrix precursor.
 7. The method asclaimed in claim 6, wherein a calcination is carried out at temperaturesfrom 450 to 700° C. within a time interval of 2 to 12 h.
 8. The methodas claimed in claim 2, wherein the matrix precursor comprises at leastone compound which is selected from the group consisting of: (i) Metalalkoxides of the formula M^(n+)(⁻OR)_(n), where R is a branched orunbranched hydrocarbon group with 1 to 12 carbon atoms and M is a metalwhich is chosen from the groups Ilb, IIIa, IIIb, IVa, IVb and VIIIb inthe periodic table of elements; and (ii) Metal halides and nitrates ofthe formula M^(n+)(X⁻)_(n), wherein X is a halide ion chosen from F⁻,Cr⁻, Br⁻ and I⁻ or a nitrate ion (NO₃ ⁻), and M is a metal chosen fromthe groups IIb, IIIa, IIIb, IVa, IVb and VIIIb in the periodic table ofelements.
 9. The method as claimed in claim 8, wherein the matrixprecursor comprises at least one compound which is selected from thegroup consisting of titanium isopropoxide, aluminum chloride, aluminumnitrate, iron (III) chloride and iron (III) nitrate.
 10. The method asclaimed in claim 2, wherein the matrix precursor has added to it atleast one solvent which is selected from the group consisting ofmethanol, ethanol, 1-propanol, 2-propanol, 1-butanol, 2-butanol and oftert-butanol.
 11. The method as claimed in claim 2, with wherein themolding has an opal structure.
 12. The method as claimed in claim 2,wherein the molding is composed of polymers or inorganic oxides whichare selected from the group of polystyrene, polymethylmethacrylate(PMMA), polydivinylbenzene, poly(styrene-co-divinylbenzene), melamineresins and silicon dioxide.
 13. A photonic crystal obtained by a methodas claimed in claim 2, characterized wherein this crystal: has apredominately convex structure, and has a bandgap or pseudo bandgapbetween a fifth and sixth bands thereof, and/or has a bandgap or pseudobandgap between a eighth and ninth bands thereof.
 14. The photoniccrystal as claimed in claim 1, said crystal having two or more bandgapsat the same time.
 15. The photonic crystal as claimed in claim 1, saidcrystal comprising cylindrical piece elements linked to one another. 16.(canceled)
 17. The photonic crystal as claimed in claim 13, said crystalhaving two or more bandgaps at the same time.
 18. A laser resonatorcomprising the photonic crystal as claimed in claim
 1. 19. A laserresonator comprising the photonic crystal as claimed in claim
 13. 20. Amatrix for optical waveguides comprising the photonic crystal as claimedin claim
 1. 21. A matrix for optical waveguides comprising the photoniccrystal as claimed in claim
 13. 22. An opalescent pigment comprising thephotonic crystal as claimed in claim
 1. 23. An opalescent pigmentcomprising the photonic crystal as claimed in claim
 13. 24. A beamsplitter comprising the photonic crystal as claimed in claim
 1. 25. Abeam splitter comprising the photonic crystal as claimed in claim 13.26. A spectral filter comprising the photonic crystal as claimed inclaim
 1. 27. A spectral filter comprising the photonic crystal asclaimed in claim 13.